So
let us see here the Clausius inequality

First we need to know the importance of Clausius
inequality in the field of thermodynamics and after that we will see here the
basic concept of Clausius inequality. We will also derive the expression for
Clausius inequality in this post.

Clausius inequality suggests us the criterion for
reversibility and irreversibility of a thermodynamic cycle.

Let us consider the following figure, we can see
here the curve between temperature (T) and Volume (V) here. There are few
almost vertically drawn lines and these lines indicate the reversible adiabatic
lines as shown and mentioned in figure. There is one more horizontal line drawn
below represent the constant temperature line and having temperature T2
as mentioned in figure.

Let us consider the cycle ABCD as shown in figure,
let us consider one process AB is one general process which will be either
reversible or irreversible process. Other processes of cycle such as process BC
and AD are reversible adiabatic process and process CD is reversible isothermal
process.

Now let us focus over the process AB and we will
clearly observe here that addition of heat will not be done here at constant
temperature but also heat will be added at different temperature during the
process AB.

Let we have divided the cycle ABCD in to number of
elementary cycles by drawing the straight lines at top as shown in figure and
these straight lines represent the constant temperature lines.

Now let us consider one elementary cycle and we can
say here that dQ heat is added here at temperature T and dQ2 heat is
rejected at temperature T2 as shown in figure. Now we will see here
the efficiency of this elementary cycle and it will be as mentioned here.

η
= 1-(dQ2/dQ)

As we know that efficiency of a general cycle will
always be less than or equal to the efficiency of a reversible cycle and
therefore we will have following equation as mentioned here

Or we can also say that for any process reversible
or irreversible such as process AB in our example, we will have following
equation

As we know that for a reversible process, we will
have

dS
= dQrev/T = dQ2/T2

For any process reversible or irreversible such as
process AB in our example, we will have following equation

The above equation will be termed as Clausius
inequality and it will suggest us the criterion for reversibility and
irreversibility of a thermodynamic cycle.